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136 VII SPACES OF CONTINUOUS FUNCTIONS

of indices iz, > m such that, for every x e E, there is at least one index z for which
IK*) - u_ml(x)\\ ^ e (use (a) and the Borel-Lebesgue axiom).

2. For any integer n > 0, let g_n be the continuous function defined in R by the conditions
that g_n(t) = 0 for r<0 and / 5* 2/n, ^(1/n) = 1, and #,(*) has the form act + ft (with
suitable constants a, /?) in each of the intervals [0, l//z] and [l//i, 2//i]. The sequence (#„)
converges simply to 0 in R, but the convergence is not uniform in any interval con-
taining / = 0.

Let m-*r_m be a bijection of N onto the set Q of rational numbers, and let

f_n(t) = £ 2~^mg_n(t — r_m). The functions/„ are continuous (7.2.1), and the sequence (/„)

converges simply to 0 in R, but the convergence is not uniform in any interval of R.

3. Let I be a compact interval of R, and (/„) a sequence of monotone real functions defined
in I, which converge simply in I to a continuous function/. Show that/is monotone,
and that the sequence (/„) converges uniformly to/in I.

4. Let E be a metric space, F a Banach space, A a dense subset of E. Let (/„) be a sequence
of bounded continuous mappings of E into F such that the restrictions of the functions
/„ to A form a uniformly convergent sequence; show that (/„) is uniformly convergent
in E.

5. Let E be a metric space, F a normed space. Show that the mapping (*, u) -» u(x) of
E x ^F°(E) into F is continuous.

6. Let E, E' be two metric spaces, F a normed space. For each mapping/of E x E' into
F and each y e E', let/, be the mapping x -»/(*, y) of E into F.

(a) Show that if/is bounded, if each f_y is continuous in E and if the mapping y ~^f_yof E' into ^F(E) is continuous, then/is continuous. Prove the converse if in addition
E is compact (use Problem 3(a) in Section 3.20).

(b) Take E = E' = F = R, and let f(x_t y) — sin xy, which is continuous and bounded
in E x E'; show that the mapping y-+f_y of E' into #F(E) is not continuous at any
point of E'.

(c) Suppose both E and E' are compact, and for any / e #_F(E x E'), let / be the
mapping y-*f_y of E' into #V(E); show that the mapping /->/ is a linear isometry of
^_F(E x EO onto ^_F<_E)(E').

7. Let E be a metric space, F a normed space. For each bounded continuous mapping /
of E into F, let G(/) be the graph of/in the space E X F.

(a) Show that /->• G(/) is a uniformly continuous injective mapping of the normed
space ^F (E) into the space §(E x F) of closed sets in E x F, which is made into a
metric space by the Hausdorff distance (see Section 3.16, Problem 3). Let T be the
image of *f?(E) by the mapping/-* G(/).

(b) Show that if E is compact, the inverse mapping G "^J of F onto ^'F (E) is continuous
(give an indirect proof).

8. Let E be a metric space with a bounded distance d. For each xeE let d_x be the bounded
continuous mapping y -*d(x,y) of E into R. Show that x-*d_x is an isometry of E
onto a subspace of the Banach space ^R(E).

3. THE STONE-WEIERSTRASS APPROXIMATION THEOREM

For any metric space E, the vector space «£(E) (resp. #£(E)) is an algebra
over the real (resp. complex) field ; from (7.1 .1) it follows that we have in that
algebra \\fg\\ < ||/|| • \\g\\, hence, by (5.5.1), the bilinear mapping (f,g)-+fg



	136 VII SPACES OF CONTINUOUS FUNCTIONS of indices iz, > m such that, for every x e E, there is at least one index z for which IK) - u_ml(x)\\ ^ e* (use (a) and the Borel-Lebesgue axiom). 2. For any integer n > 0, let g_n be the continuous function defined in R by the conditions that g_n(t) = 0 for r<0 and / 5* 2/n, ^(1/n) = 1, and #,() has the form act* + ft (with suitable constants a, /?) in each of the intervals [0, l//z] and [l//i, 2//i]. The sequence (#„) converges simply to 0 in R, but the convergence is not uniform in any interval con- taining / = 0. Let m-r_m* be a bijection of N onto the set Q of rational numbers, and let oc f_n(t) = £ 2~^mg_n(t — r_m). The functions/„ are continuous (7.2.1), and the sequence (/„) converges simply to 0 in R, but the convergence is not uniform in any interval of R. 3. Let I be a compact interval of R, and (/„) a sequence of monotone real functions defined in I, which converge simply in I to a continuous function/. Show that/is monotone, and that the sequence (/„) converges uniformly to/in I. 4. Let E be a metric space, F a Banach space, A a dense subset of E. Let (/„) be a sequence of bounded continuous mappings of E into F such that the restrictions of the functions /„ to A form a uniformly convergent sequence; show that (/„) is uniformly convergent in E. 5. Let E be a metric space, F a normed space. Show that the mapping (, u) -» u(x)* of E x ^F°(E) into F is continuous. 6. Let E, E' be two metric spaces, F a normed space. For each mapping/of E x E' into F and each y e E', let/, be the mapping x -»/(, y) of E into F. (a) Show that if/is bounded, if each f_y* is continuous in E and if the mapping y ~^f_yof E' into ^F(E) is continuous, then/is continuous. Prove the converse if in addition E is compact (use Problem 3(a) in Section 3.20). (b) Take E = E' = F = R, and let f(x_t y) — sin xy, which is continuous and bounded in E x E'; show that the mapping y-+f_y of E' into #F(E) is not continuous at any point of E'. (c) Suppose both E and E' are compact, and for any / e #_F(E x E'), let / be the mapping y-f_y* of E' into #V(E); show that the mapping /->/ is a linear isometry of ^_F(E x EO onto ^_F<_E)(E'). 7. Let E be a metric space, F a normed space. For each bounded continuous mapping / of E into F, let G(/) be the graph of/in the space E X F. (a) Show that /->• G(/) is a uniformly continuous injective mapping of the normed space ^F (E) into the space §(E x F) of closed sets in E x F, which is made into a metric space by the Hausdorff distance (see Section 3.16, Problem 3). Let T be the image of f?(E) by the mapping/- G(/). (b) Show that if E is compact, the inverse mapping G "^J of F onto ^'F (E) is continuous (give an indirect proof). (c) Show that if E = [0,1] and F = R, G"¹ is not uniformly continuous. 8. Let E be a metric space with a bounded distance d. For each xeE let d_x be the bounded continuous mapping y -d(x,y)* of E into R. Show that x-d_x* is an isometry of E onto a subspace of the Banach space ^R(E).

	3. THE STONE-WEIERSTRASS APPROXIMATION THEOREM

	For any metric space E, the vector space «£(E) (resp. #£(E)) is an algebra over the real (resp. complex) field ; from (7.1 .1) it follows that we have in that algebra \\fg\\ < \|\|/\|\| • \\g\\, hence, by (5.5.1), the bilinear mapping (f,g)-+fg